| Linear Programming |
Article Index for Linear |
Website Links For Linear Programming |
Information AboutLinear Programming |
| CATEGORIES ABOUT LINEAR PROGRAMMING | |
| optimization | |
| operations research | |
|
Linear programming is an important field of optimization for several reasons. Many practical problems in Operations Research can be expressed as linear programming problems. Certain special cases of linear programming, such as ''network flow'' problems and ''multicommodity flow'' problems are considered important enough to have generated much research on specialized algorithms for their solution. A number of algorithms for other types of optimization problems work by solving LP problems as sub-problems. Historically, ideas from linear programming have inspired many of the central concepts of optimization theory, such as ''duality,'' ''decomposition,'' and the importance of ''convexity'' and its generalizations. STANDARD FORM ''Standard form'' is the usual and most intuitive form of describing a linear programming problem. It consists of the following three parts:
: e.g. maximize
: e.g. :: ::
: e.g. :: The problem is usually expressed in '' Matrix form'', and then becomes: : maximize : subject to Other forms, such as minimization problems, problems with constraints on alternative forms, as well as problems involving negative Variable s can always be rewritten into an equivalent problem in standard form. Example Suppose that a farmer has a piece of farm land, say ''A'' square kilometres large, to be planted with either wheat or barley or some combination of the two. The farmer has a limited permissible amount ''F'' of fertilizer and ''P'' of insecticide which can be used, each of which is required in different amounts per unit area for wheat (''F''1, ''P''1) and barley (''F''2, ''P''2). Let ''S''1 be the selling price of wheat, and ''S''2 the price of barley. If we denote the area planted with wheat and barley with ''x''1 and ''x''2 respectively, then the optimal number of square kilometres to plant with wheat vs barley can be expressed as a linear programming problem: Which in matrix form becomes: : maximize : subject to AUGMENTED FORM (SLACK FORM) Linear programming problems must be converted into ''augmented form'' before being solved by the Simplex Algorithm . This form introduces non-negative ''slack variables'' to replace non-equalities with equalities in the constraints. The problem can then be written on the following form: : Maximize ''Z'' in: : : where ''xs'' are the newly introduced slack variables, and ''Z'' is the variable to be maximized. Example The example above becomes as follows when converted into augmented form: where are (non-negative) slack variables. Which in matrix form becomes: : Maximize ''Z'' in: : DUALITY Every linear programming problem, referred to as a primal problem, can be converted into an equivalent dual problem. In matrix form, we can express the ''primal problem'' as: : maximize : subject to The equivalent ''dual problem'' is: : minimize : subject to where ''y'' is used instead of ''x'' as variable vector. THEORY Geometrically, the linear constraints define a Convex Polyhedron , which is called the ''feasible region''. Since the objective function is also linear, all local optima are automatically global optima. The linear objective function also implies that an optimal solution can only occur at a boundary point of the feasible region. There are two situations in which no optimal solution can be found. First, if the constraints contradict each other (for instance, ''x'' ≥ 2 and ''x'' ≤ 1) then the feasible region is empty and there can be no optimal solution, since there are no solutions at all. In this case, the LP is said to be ''infeasible''. Alternatively, the Polyhedron can be unbounded in the direction of the objective function (for example: maximize ''x''1 + 3 ''x''2 subject to ''x''1 ≥ 0, ''x''2 ≥ 0, ''x''1 + ''x''2 ≥ 10), in which case there is no optimal solution since solutions with arbitrarily high values of the objective function can be constructed. Barring these two pathological conditions (which are often ruled out by resource constraints integral to the problem being represented, as above), the optimum is always attained at a vertex of the polyhedron. However, the optimum is not necessarily unique: it is possible to have a set of optimal solutions covering an edge or face of the polyhedron, or even the entire polyhedron (This last situation would occur if the objective function were constant). ALGORITHMS .]] The problem was in Complexity Class NP , or if it was solvable in Polynomial Time (complexity class P ). The first worst-case polynomial-time algorithm for the linear programming problem was proposed by Leonid Khachiyan in 1979 . It was based on the Ellipsoid Method in Nonlinear Optimization by Naum Shor , which is the generalization of the ellipsoid method in Convex Optimization by Arkadi Nemirovski , a 2003 John Von Neumann Theory Prize winner, and D. Yudin . However, the practical performance of Khachiyan's algorithm is disappointing: generally, the simplex method is more efficient. Its main importance is that it encouraged the research of Interior Point Method s. In contrast to the simplex algorithm, which only progresses along points on the boundary of the feasible region, interior point methods can move through the interior of the feasible region. In , which performs very well in practice even though little is known about it theoretically. The current opinion is that the efficiency of good implementations of simplex-based methods and interior point methods is similar for routine applications of linear programming. LP solvers are in widespread use for optimization of various problems in industry, such as optimization of flow in transportation networks, many of which can be transformed into linear programming problems only with some difficulty. Open problems
INTEGER UNKNOWNS If the unknown variables are all required to be integers, then the problem is called an integer programming (IP) or '''integer linear programming''' (ILP) problem. In contrast to linear programming, which can be solved efficiently in the worst case, integer programming problems are in the worst case undecidable, and in many practical situations (those with bounded variables) NP-hard . '''0-1 integer programming''' is the special case of integer programming where variables are required to be 0 or 1 (rather than arbitrary integers). This method is also classified as NP-hard , and in fact the decision version was one of Karp's 21 NP-complete Problems . If only some of the unknown variables are required to be integers, then the problem is called a mixed integer programming (MIP) problem. These are generally also NP-hard . There are however some important subclasses of IP and MIP problems that are efficiently solvable, most notably problems where the constraint matrix is Totally Unimodular and the right-hand sides of the constraints are integer. Advanced algorithms for solving integer linear programs include:
SEE ALSO
;Solver packages
REFERENCES
EXTERNAL LINKS
;Software
|
|
|